 So let's do a little short video on what the probability distribution is for rolling two six-sided die, okay? And we're gonna come back to this a lot further when we do a series on craps, which is gonna be basically our introduction to probability and statistics, okay? But we're at least a year, two years away from that. And I wanted to make this video just to show you guys what the distribution is because it connects back to series four, units and ratios, what we're talking about right now. And once we finish series four, which is basically units and ratios, graphs and functions, and a little bit more on zero infinity, we'll probably go directly into probability statistics, which is basically, I'd consider this to this video to be a primer for that, okay? So what we're going to do is take a look at what the probability distribution is of rolling two six-sided die are, right? When it comes to two dice like this, the smallest number you can get is a two, basically a one on the first die and a one on the second die, okay? That's the smallest number you can get. And the biggest number you can get is a 12, right? A six on the first die and a six on the second die. So those are our limits of the graph that we're gonna put up. And what we're gonna do is create a little bar graph with the possible roll outcomes on the x-axis and the probability, the distribution of the different combinations of how we can get a specific number on the y-axis. So like we said, the lowest number that you can get is a two, right? And the highest number you can get is a 12. And in between is basically the counting numbers, right? So it goes 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. And on the y-axis, what we're gonna do is put the possible outcomes, right? The different ways we can get each one of those numbers. And the possible outcomes that we can have, the maximum possible outcome is with the seven, which there are six ways, and we'll talk about this. And the lowest possible outcome we can have is one way to get a two or a 12, okay? So the way that this works is for number two, if you wanna get a two, there's only one way you can get a two, okay? You need a one out of six on the first one. You need to get the number one on the first one, and you need to get the number one on the second die, right? And when it comes to probabilities like that, it's pretty straightforward. What you do, you multiply the two probabilities together. So you need a one out of six on the first die, right? So that's one out of six times one out of six for the second die, which basically gives us one out of 36. The probability of getting a two with a two six sided die is one out of 36. And there's only one combination to this. Let's take a look at number three, right? To be able to get the number three, you can either get a one on the first die or a two on the first die. If you get a one on the first die, you need to get a two on the second die. If you get a two on the first die, you need to get a one on the second die. The probability for this, to figure out the probability for this, you go two out of six, which is you can get two different numbers on the first die and still make a three with a second die. So it'd be two out of six times one out of six. Because no matter what you get on the first die, you need to get a specific number on the second die. So for example, if you get a one on the first die, you can't get another one on the second die, you need to get a two, right? If you get a two on the first die, you need to get a one on the second die. So your probability reduces to only one choice on the second die and two choices on the first die. So two out of six times one out of six is two out of 36. And that's the probability of getting a three. And how many combinations do we have? We have two possible combinations, a one and a two and a two and a one. Two get a number four, okay. You can get a one, a two, or a three on the first die. And on the second die, you have to match. So if you get a one on the first die, you need to get a three on the second die. If you get a two on the first die, you need to get a two on the second die. Where is it? A two on the second die. If you get a three on the first die, you need to get a one on the second die. So there's three possible combinations. And the probability distribution would be, there's three choices for the first die and only one choice. You have to match it for the second, with the second die, right? So there's three possible combination and the probability distribution would be, the probability would be three out of 36 times, three out of six times one out of six, which is three out of 36 total, right? For number five possible combinations, one, two, three, or four on the first die and you match it with the other die, right? If you get a one on the first die, you get a four on the second die and so on and so forth. So there is four possible combinations, possible ways to roll a five. And the probability for it is four out of six times one out of six, which is four out of 36. For number six, there's five different combinations. For number seven, there's six different combinations. So the way this works is seven has the most possible combinations of any of the other numbers. Because no matter what you roll on the first die, you can always make a seven with the second die. If you roll a one on the first die, the possible combinations, the possible numbers that you can roll are two, three, four, five, six, or seven, right? You can't roll an eight. Rates eight and above are not possible. If you get a one on the first die, if you get a six on the first die, then your only choices are seven, eight, nine, ten, eleven, twelve, right? And it is 100% symmetrical. The symmetry is around the seven and the combinations, the numbers going away from seven are exactly the same way as these guys, right? So if you have the seven here, six have five possible outcomes. Number eight has five possible outcomes. If you go two units away from seven, you're at the number five. That's the same thing as number nine, right? So the graph comes up, goes down, perfect symmetry. And this is what the probability distribution looks like for rolling two dies, a bar graph. One combination here, two, three, four, five, six combination. Seven has the most possible combinations of getting. And then you go back down again, five, four, three, two, one, okay? Now there's another beautiful symmetry with dice, which is embedded in the design of the dios. And that's basically the number that you see on the front of your roll. That plus the back number, whatever's on the back, adds up to seven. So if you get a six on the front here, the back is one. If you get a three, the back is four, right? If you get a five, the back is two, right? So the other symmetry that's centered around seven is whatever number you get on the front. The back adds up to make it seven. So with two dice, no matter what you have in the front, if it's a six, on the back would be an eight to give you seven. So if you get a six on the front, you see six on the front. On the back is the eight. If you see five on the front, on the back is a nine. If you see four on the front, on the back is a 10. Three as 11, two is 12, right? And on the back of seven is seven. So what we just did was basically do a quick little primer for what the probability distribution is for dice, for two six-dice dice. The reason being, we're in the units and ratios section, and this comes into play. So we're going to refer back to this video. In future videos, and we're basically going to build on this. When we get into a series on craps, which is going to lead us directly to probability and statistics, once we finish series four. And that's going to be in 2015 or 2016. I'm not 100% sure what I'm going to get to it, right? But this is basically it. I hope you enjoyed. If you've got any questions, post a comment, send your message. We'll talk about this more if you want. Okay, bye-bye.